I am now reading this wonderful thesis on moduli spaces by Djounvona.
On Proposition 2.5.3 the author says that if you have a moduli functor $\mathcal{M}$ that is represented by an object $M$ in a small category $\mathcal{C}$, then for each family $X/B$ of objects of $\mathcal{M}$ parametrized by each $B$ in $\mathcal{C}$ then there exists a morphism $\varphi : B \to M$.
I can understand how one can set-theoretically define this map, but in any way can see why this have the structure of a morphim of the category $\mathcal{C}$.
Any kind of help will be useful.