Let $X\colon \mathsf{C^{op}}\to \mathsf{Set}$ be a presheaf. It's category of elements, denoted by $\int X$, has pairs $(a,s)$ with $s \in X(a)$ as objects and $f \in \mathrm{Hom}_{\mathsf{D}}(a,b)$ such that $X(f)(t) = s$ as morphisms $(a,s)\to (b,t)$.
This is a theorem from the book Higher Categories and Homotopical Algebra by D.C.Cisinski.
Let $\mathsf{A}$ be a small category, together with a locally small category $\mathsf{C}$ which admits small colimits. For any functor $u\colon \mathsf{A}\to \mathsf{C}$, the functor of evaluation at $u$ $$u^*\colon \mathsf{C}\to\widehat{\mathsf{A}}, Y \mapsto u^*(Y) = (a\mapsto\mathrm{Hom}_{\mathsf{C}}(u(a),Y))$$ has a left adjoint $$u_{!}\colon\widehat{\mathsf{A}}\to\mathsf{C}.$$ Moreover, there is a natural isomorphism $u(a) \cong u_{!}(\mathrm{Hom}_{\mathsf{D}}(-,a)), a \in \mathsf{A}$, such that, for any object $Y$ of $\mathsf{C}$, the induced bijection $\mathrm{Hom}_{\mathsf{C}}(u_{!}(\mathrm{Hom}_{\mathsf{D}}(-,a)),Y) \cong \mathrm{Hom}_{\mathsf{C}}(u(a),Y)$ is the inverse of the composition of the Yoneda bijection $\mathrm{Hom}_{\mathsf{C}}(u(a),Y) = u^*(Y)_a = \mathrm{Hom}_{\widehat{\mathsf{A}}}(\mathrm{Hom}_{\mathsf{D}}(-,a),u^*(Y))$ with the adjunction formula $\mathrm{Hom}_{\widehat{\mathsf{A}}}(\mathrm{Hom}_{\mathsf{D}}(-,a),u^*(Y)) \cong \mathrm{Hom}_{\mathsf{C}}(u_{!}(\mathrm{Hom}_{\mathsf{D}}(-,a)),Y)$.
Cisinski constructs a left adjoint functor $u_!$ by setting, for each presheaf $X$ over $\mathsf{A}$, $u_!(X)$ to be a colimit of the functor $F\colon\int X\to \mathsf{C}$ such that $F(a,s) = u(a)$ (I assume that $F(f\colon (a,s)\to (b,t)) = u(f)$, but the author doesn't explicitly state this, so it is possible I'm mistaken).
It is also stated what is the action of $u_!$ on morphisms of presheaves, and I can't guess it, though it is crucial to the rest of the proof as we need to prove naturality of an adjunction formula. This is my first question.
My second question is why $u(a) \cong u_!(\mathrm{Hom}_{\mathsf{D}}(-,a))$ is unique. Sure, the inverse of a bijection is unique, so the induced bijection $\mathrm{Hom}_{\mathsf{C}}(u_{!}(\mathrm{Hom}_{\mathsf{D}}(-,a)),Y) \cong \mathrm{Hom}_{\mathsf{C}}(u(a),Y)$ is unique, but this doesn't explain why $u(a) \cong u_{!}(\mathrm{Hom}_{\mathsf{D}}(-,a))$ is.