In Rosebrugh and Wood's paper Pullback preserving functors, there is a coend calculation that is mostly straightforward, but contains one step I do not follow. I suspect the context is not important for the question, but I will provide some, just in case it is.
Let $(-)_* : \mathscr L^{\text{co}} \to \mathscr N^{\text{co}}$ be a proarrow equipment, i.e. a locally full faithful pseudofunctor for which, for each 1-cell $f : X \to Y$ in $\mathscr L^{\text{co}}$, the 1-cell $f_*$ admits a right adjoint $f^*$ in $\mathscr N^{\text{co}}$. Furthermore, let $p : S \to M \leftarrow T : q$ be a codiscrete cofibration in $\mathscr L^{\text{co}}$.
In Lemma 3.2 of Rosebrugh and Wood's paper, it is proven (in lines 4 – 5 of the calculation) that, given $f \in \mathscr L(S, X)$, $g \in \mathscr L(T, X)$, and $h \in \mathscr L(M, X)$,
\begin{equation} \int^{h \in \mathscr L(M, X)} \mathscr N(M, X)(h, p^* f) \times \mathscr N(T, X)(g, q h) \cong \mathscr N(T, X)(g, q p^* f) \tag{A} \end{equation}
This looks like an application of the Yoneda lemma/density formula, in which we substitute $p^* f$ for $h$. However, $h$ ranges over objects in $\mathscr L(M, X)$, not objects of $\mathscr N(M, X)$, so I do not see that the Yoneda lemma applies directly.
Why does the step $(\text{A})$ follow?
(Note that I have corrected a typo in the original paper, in which $\mathscr N(S, X)$ appears instead of $\mathscr N(M, X)$.)
