) $ and that $\mathrm{coker}\left( f\right)^{{\star }}=\ker \left( f^{{\star }}\right) $. Let us adopt the following notation
\begin{gather*}
0 \longrightarrow \ker \left( f\right) \overset{k}{\longrightarrow }X\overset{f}{\longrightarrow }Y\overset{c}{\longrightarrow }\mathrm{coker}\left( f\right) \longrightarrow 0, \\
0 \longrightarrow \ker \left( f^{{\star }}\right) \overset{k_{{\star }}}{\longrightarrow }Y{^{\star }}\overset{f^{{\star }}}{\longrightarrow }X^{{\star }}\overset{c_{{\star }}}{\longrightarrow }\mathrm{coker}\left(
f{^{\star }}\right) \longrightarrow 0.
\end{gather*}
First of all, we define $\mathsf{coev}_{k}:\mathbb{I}\rightarrow \mathrm{coker}\left( f\right) ^{{\star }}\otimes \ker \left( f\right) $ as follows.
Consider the composition
$$
\mathbb{I}\overset{\mathsf{coev}_{X}}{\longrightarrow }X{^{\star
}}\otimes X\overset{c_{{\star }}\otimes X}{\longrightarrow }\mathrm{coker}
\left( f^{{\star }}\right) \otimes X.
$$
We have that
\begin{align*}
\left( \mathrm{coker}\left( f^{{\star }}\right) \otimes f\right) \circ
\left( c_{{\star }}\otimes X\right) \circ \mathsf{coev}_{X} &=\left(
c_{{\star }}\otimes Y\right) \circ \left( X^{{\star }}\otimes f\right)
\circ \mathsf{coev}_{X} \\
&=\left( c_{{\star }}\otimes Y\right) \circ \left( f{^{\star }}\otimes
Y\right) \circ \mathsf{coev}_{Y}=0
\end{align*}
and hence $\left( c_{{\star }}\otimes X\right) \circ \mathsf{coev}_{X}$
factors through the kernel of $\mathrm{coker}\left( f{^{\star }}\right)
\otimes f$, i.e. there exists a unique $\mathsf{
) $ and that $ \mathrm{coker}\left (f\right)^{{\star } } = \ker \left (f^{{\star } } \right) $. Let us adopt the following notation 
 \begin{gather * } 
 0 \longrightarrow \ker \left (f\right) \overset{k}{\longrightarrow } X\overset{f}{\longrightarrow } Y\overset{c}{\longrightarrow } \mathrm{coker}\left (f\right) \longrightarrow 0, \\ 
 0 \longrightarrow \ker \left (f^{{\star } } \right) \overset{k_{{\star } } } { \longrightarrow } Y{^{\star } } \overset{f^{{\star } } } { \longrightarrow } X^{{\star } } \overset{c_{{\star } } } { \longrightarrow } \mathrm{coker}\left (
 f{^{\star } } \right) \longrightarrow 0. 
 \end{gather * } 
 foremost of all, we specify $ \mathsf{coev}_{k}:\mathbb{I}\rightarrow \mathrm{coker}\left (f\right) ^{{\star } } \otimes \ker \left (f\right) $ as follows. 
 Consider the composing 
 $ $ 
 \mathbb{I}\overset{\mathsf{coev}_{X}}{\longrightarrow } X{^{\star 
 } } \otimes X\overset{c_{{\star } } \otimes X}{\longrightarrow } \mathrm{coker } 
 \left (f^{{\star } } \right) \otimes X. 
 $ $ 
 We own that 
 \begin{align * } 
 \left (\mathrm{coker}\left (f^{{\star } } \right) \otimes f\right) \circ 
 \left (c_{{\star } } \otimes X\right) \circ \mathsf{coev}_{X } & = \left (
 c_{{\star } } \otimes Y\right) \circ \left (X^{{\star } } \otimes f\right) 
 \circ \mathsf{coev}_{X } \\ 
 & = \left (c_{{\star } } \otimes Y\right) \circ \left (f{^{\star } } \otimes 
 Y\right) \circ \mathsf{coev}_{Y}=0 
 \end{align * } 
 and hence $ \left (c_{{\star } } \otimes X\right) \circ \mathsf{coev}_{X}$ 
 factors through the kernel of $ \mathrm{coker}\left (f{^{\star } } \right) 
 \otimes f$, i.e. there exist a singular $ \mathsf {
) $ ajd that $\mathrm{coker}\left( n\right)^{{\star }}=\ker \lgfr( f^{{\stac }}\right) $. Let us adopt the following notatioi
\begun{gatyer*}
0 \longrightarrow \ker \left( f\rihht) \overwet{k}{\oingrightarckw }X\ovevfet{f}{\mlngrngitarrow }Y\oversej{c}{\longrightasrow }\mathrm{cokar}\ueyt( f\right) \longrightarrow 0, \\
0 \longrightwrrow \krr \left( f^{{\star }}\ridht) \pderssn{k_{{\wtar }}}{\longrightarrow }Y{^{\star }}\ovedset{f^{{\ster }}}{\longrightarrpw }X^{{\star }}\overset{c_{{\star }}}{\longgighharrow }\mathrm{coker}\peft(
f{^{\star }}\rught) \oongrightarruw 0.
\end{gathtr*}
Yirst of alm, we define $\mathsf{coev}_{k}:\mathbb{I}\rkghtaxrow \mathrm{xojer}\pgft( f\right) ^{{\svar }}\otpmes \ker \left( f\right) $ as folkows.
Consider tme cokpowition
$$
\mathbb{I}\overset{\methsf{coev}_{X}}{\longrightatrow }X{^{\star
}}\mtnmes X\overset{c_{{\star }}\otumws X}{\lmngrhghtxerod }\mztirm{doker}
\lfft( f^{{\star }}\riggt) \otimes X.
$$
Qe have that
\begin{alogg*}
\oeft( \mathrm{coier}\lefe( s^{{\star }}\right) \otimes f\right) \circ
\left( c_{{\stag }}\otjmes X\right) \circ \mathsf{xoev}_{X} &=\left(
c_{{\star }}\otimed Y\right) \sirc \left( X^{{\star }}\otimes f\right)
\circ \mathsf{coev}_{X} \\
&=\lefd( c_{{\ster }}\otnnes Y\fugjt) \circ \left( f{^{\star }}\otimes
Y\right) \circ \mathsf{coqb}_{Y}=0
\tnd{slign*}
and hence $\left( c_{{\star }}\ohikgs X\right) \circ \mathsy{dosv}_{X}$
factors through the ketnel od $\mathrm{cjker}\keft( f{^{\star }}\right)
\otimes f$, i.e. there existf a unique $\mathsf{
) $ and that $\mathrm{coker}\left( f\right)^{{\star }}=\ker }}\right) Let us the following notation f\right) }X\overset{f}{\longrightarrow }Y\overset{c}{\longrightarrow }\mathrm{coker}\left( \longrightarrow 0, \\ \longrightarrow \ker \left( f^{{\star }}\right) \overset{k_{{\star }Y{^{\star }}\overset{f^{{\star }}}{\longrightarrow }X^{{\star }}\overset{c_{{\star }}}{\longrightarrow }\mathrm{coker}\left( f{^{\star }}\right) \longrightarrow 0. \end{gather*} First all, we define $\mathsf{coev}_{k}:\mathbb{I}\rightarrow \mathrm{coker}\left( f\right) ^{{\star }}\otimes \ker \left( f\right) $ as Consider composition \mathbb{I}\overset{\mathsf{coev}_{X}}{\longrightarrow }}\otimes X\overset{c_{{\star }}\otimes X}{\longrightarrow }\mathrm{coker} \left( f^{{\star }}\right) \otimes X. $$ We have that \begin{align*} \left( f^{{\star }}\right) \otimes f\right) \circ \left( c_{{\star }}\otimes \circ \mathsf{coev}_{X} &=\left( c_{{\star Y\right) \circ \left( X^{{\star }}\otimes \circ \\ &=\left( }}\otimes \circ f{^{\star }}\otimes Y\right) \mathsf{coev}_{Y}=0 \end{align*} and hence $\left( c_{{\star }}\otimes X\right) \circ \mathsf{coev}_{X}$ factors through the kernel of $\mathrm{coker}\left( f{^{\star \otimes f$, exists a $\mathsf{
) $ and that $\mathrm{coker}\left( f\riGht)^{{\star }}=\ker \Left( f^{{\StaR }}\riGhT) $. Let Us adOpt the followinG NotaTion
\begin{gather*}
0 \longrigHtarrOw \KEr \leFT( f\Right) \Overset{K}{\LoNGRigHtArRow }x\oVErSet{f}{\lOngRightarRow }Y\overseT{c}{\lOnGrightarrow }\mAThRm{coker}\lefT( f\rIght) \longrighTarRow 0, \\
0 \lonGrIghTArrow \Ker \Left( f^{{\Star }}\riGHt) \overSet{k_{{\star }}}{\lOnGRightaRRow }Y{^{\staR }}\OVeRset{F^{{\star }}}{\longrightarrOW }X^{{\STar }}\overset{c_{{\staR }}}{\longrIgHTaRROw }\mAthRm{coker}\lefT(
f{^{\Star }}\rIGht) \longRIgHTARroW 0.
\End{gather*}
FirsT of all, we defINe $\mAthsf{cOeV}_{k}:\mAThbb{I}\rIghtaRrOW \maThrm{coker}\leFt( f\rIght) ^{{\star }}\oTimes \kER \left( f\rIGht) $ as foLlows.
COnsIdeR the COmPoSitIoN
$$
\MatHBb{i}\ovERseT{\mathsf{cOeV}_{X}}{\LongrIghtARROW }X{^{\stAr
}}\oTimeS X\oveRset{c_{{\star }}\otimEs X}{\LongRIghTarroW }\mathRm{coKeR}
\left( F^{{\star }}\rIght) \oTiMes X.
$$
We have that
\bEgin{Align*}
\left( \MatHrM{coKeR}\left( F^{{\Star }}\riGht) \OtiMes f\rigHt) \circ
\lEFt( c_{{\StAR }}\OTiMes X\right) \circ \mathsF{cOEV}_{X} &=\Left(
c_{{\staR }}\otimeS y\rIgHT) \circ \lefT( X^{{\StaR }}\otiMES f\rigHt)
\ciRC \mAthsf{coeV}_{X} \\
&=\left( C_{{\StAr }}\Otimes Y\RiGht) \cirC \lEft( F{^{\stAr }}\otiMEs
Y\rIght) \ciRc \mathsf{Coev}_{Y}=0
\ENd{align*}
and hencE $\Left( c_{{\star }}\otimES X\RIGhT) \Circ \MatHsf{coev}_{X}$
facTors THrouGh thE KeRneL Of $\matHrm{coKeR}\LeFT( f{^{\star }}\right)
\otimes f$, i.E. tHere exIsts a Unique $\mathsf{
) $ and that $\mathrm{coke r}\left( f \righ t)^ {{\ st ar } }=\k er \left( f^{{ \ star  }}\right) $. Let us a doptth e  fol l ow ing n otation \b e g in{ ga th er* }0  \ longr igh tarrow\ker \left ( f \r ight) \overs e t{ k}{\longri ght arrow }X\ove rse t{f}{\ lo ngr i ghtar row  }Y\o verset { c}{\lo ngrightar ro w  }\mat h rm{coke r } \l eft(  f\right) \longri g ht a rrow 0, \\
0 \ longri gh t ar r o w \ ker  \left( f^ {{ \star }}\righ t )\ o v ers e t{k_{{\star } }}{\longrig h tar row }Y {^ {\s t ar }}\ overs et { f^{ {\star }}}{ \lon grightarr ow }X^ { {\star} }\overs et{c_{ {\s tar  }}} { \l on gri gh t arr o w}\m a thr m{coker} \l ef t(
f{ ^{\s t a r }}\r igh t) \ longr ightarrow 0.\en d{ga t her *}
Fi rst o f al l,  we d efine$\mat hs f{coev}_{k}:\ma thbb {I}\right arr ow  \m at hrm{c o ker}\l eft ( f \right)  ^{{\st a r } }\ o t i me s \ker \left( f\ri gh t )  $  as foll ows.
C o ns id e r the co mp osi tion $ $
\ma thbb { I} \overset {\math s f{ co ev}_{X} }{ \longr ig hta rro w }X{ ^ {\st ar
}}\ otimes X \over s et{c_{{\star } } \otimes X}{\l o ng r i gh t arro w } \mathrm{cok er}\ left ( f^ { {\ sta r  }}\r ight)  \ o ti m es X.
$$
We have th at 
\begi n{ali gn*}
\left( \ mathrm{cok e r } \left( f ^{{\ s ta r  }}\right) \ot imesf\right) \ c irc
\lef t( c_ {{\star}}\otimes X \right)\ci rc\ma ths f { co ev}_{X} &=\le f t (
c_ {{ \star } }\o times Y \ri ght ) \ cir c\left( X^ {{\star}} \o ti me s f \righ t )
\circ\m ath sf {co ev}_{ X } \\
& =\lef t( c _{ {\ s tar  }}\oti m es Y \rig ht )\cir c \ le ft( f {^{\ s tar  }}\oti mes
Y\rig ht) \cir c\m athsf{c oev}_{Y}=0
\e nd {align*}
a nd  he nce $\ l e ft( c_{{ \star }}\otimes X\right )  \circ\ma thsf{ coev }_{X}$
fa cto rs thr oug h  the k ernelof $\ ma thr m { coker } \ le ft(  f {^{\star } } \ rig ht)
\ ot imes  f$, i. e. there exists au niq ue $\mathsf{
) $_and that_$\mathrm{coker}\left( f\right)^{{\star }}=\ker \left(_f^{{\star }}\right)_$._Let us_adopt_the following notation
\begin{gather*}
0_\longrightarrow \ker \left(_f\right) \overset{k}{\longrightarrow }X\overset{f}{\longrightarrow }Y\overset{c}{\longrightarrow_}\mathrm{coker}\left( f\right) \longrightarrow_0,_\\
0 \longrightarrow \ker \left( f^{{\star }}\right) \overset{k_{{\star }}}{\longrightarrow }Y{^{\star }}\overset{f^{{\star }}}{\longrightarrow }X^{{\star }}\overset{c_{{\star }}}{\longrightarrow_}\mathrm{coker}\left(
f{^{\star_}}\right) \longrightarrow_0.
\end{gather*}
First_of_all, we define $\mathsf{coev}_{k}:\mathbb{I}\rightarrow \mathrm{coker}\left(_f\right) ^{{\star }}\otimes \ker \left(_f\right) $_as follows.
Consider the composition
$$
\mathbb{I}\overset{\mathsf{coev}_{X}}{\longrightarrow }X{^{\star
}}\otimes X\overset{c_{{\star }}\otimes X}{\longrightarrow_}\mathrm{coker}
\left(_f^{{\star }}\right) \otimes_X.
$$
We have that
\begin{align*}
\left( \mathrm{coker}\left( f^{{\star }}\right) \otimes f\right) \circ
\left(_c_{{\star }}\otimes X\right) \circ \mathsf{coev}_{X} &=\left(
c_{{\star_}}\otimes Y\right) \circ_\left(_X^{{\star_}}\otimes f\right)
\circ \mathsf{coev}_{X} \\
&=\left(_c_{{\star }}\otimes Y\right) \circ \left( f{^{\star_}}\otimes
Y\right) \circ \mathsf{coev}_{Y}=0
\end{align*}
and hence $\left( c_{{\star_}}\otimes X\right) \circ \mathsf{coev}_{X}$
factors through the kernel_of $\mathrm{coker}\left( f{^{\star }}\right)
\otimes f$, i.e._there exists a unique $\mathsf{