The Clowder Project

Let $A,B,C,D\in \text{Obj}\webleft (\mathcal{C}\webright )$ and let $f\colon A\to B$ and $g\colon B\to C$ be morphisms of $\mathcal{C}$.

1. Interaction Between Precomposition and Postcomposition. We have
   
2. Interaction With Composition I. We have
   
3. Interaction With Composition II. We have
   
4. Interaction With Composition III. We have
   
5. Interaction With Identities. We have
   \begin{align*} \webleft (\text{id}_{A}\webright )^{*} & = \text{id}_{\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )},\\ \webleft (\text{id}_{B}\webright )_{*} & = \text{id}_{\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )}. \end{align*}