Substitution by structural recursion Following the article's notation, I write $\mathcal{F}$ for the
category of presheaves on a (suitable) category $\mathbb{F}$, $TV$ for the
presheaf of terms, $\delta$ for the context extension, and
$\bullet$ for the product of the $\mathcal{F}$-monoid.

In Abstract Syntax and Variable Binding [Fiore, Plotkin, Turi], the
authors set out to define substitution by structural recursion
(Section 3). I was expecting to see substitution expressed as an
initial algebra of some sort.

Instead, from what I understand, the authors:


 Construct the substitution $\sigma : \delta(TV) \times TV \to TV$ by some universal construction ("Definition of substitution by structural recursion") 
 Show that it forms a substitution algebra (Theorem 3.2) 
 Show that the categories of substitution algebras and clones are equivalent (Theorem 3.3) 
 Show that the categories of clones and monoids in $\mathcal{F}$ are equivalent (Proposition 3.4) 
 Show that subtitution is the multiplication of such a monoid (Proposition 3.5) 
 Conclude that $\sigma$, the substitution, is defined by structural recursion (Corollary 3.6). 


I'm failing to appreciate their motive in moving from substitution
algebras to clones to monoids. Is that a natural thing to do for a
mathematician? 

In particular, why not stay focused on substitution algebras and, I
guess, present substitution as the initial one?

Conversely, they claim that Corollary 3.6 gives a definition of
substitution by structural recursion: how is that? It is just said
that "$\sigma$ is the unique homomorphic extension of $V \bullet TV
\cong TV$": how did they derive their example (substitution for the
lambda calculus, very end of Section 3) from this statement?
A more tutorial-paced presentation of the same/related material is available in:
Roy L. Crole: Basic Category Theory for Models of Syntax. Generic Programming 2003: 133-177. This should help.