Let us consider a category $\mathfrak{A}$ and the following diagram of morphisms:

$\require{AMScd}$
\begin{CD}
A @>{f}>> B\\
@V{h}VV @V{k}VV\\
C @>{g}>> D
\end{CD}

Assume that $k \circ f=g \circ h$ and that there exists a unique $w: B \longrightarrow C$ such that $w \circ f=h$. May I also deduce that $g \circ w=k$?