Given a simplicial set $X$, denote by $(\mathrm{Set}_{\Delta})/_X$ the over category, whose objects are morphisms of simplicial sets with source $X$. I want to show that the joint $X \star Y$ of simplicial sets induces a functor \begin{align*} X \star - \colon \mathrm{Set}_{\Delta} & \to (\mathrm{Set}_{\Delta})/_X\\ Y & \mapsto X \hookrightarrow X \star Y \end{align*}
There has to be something really stupid that I'm missing, because I can't make sense of what that functor is supposed to do on morphisms. Given a morphism $p \colon Y_1 \to Y_2$ of simplicial sets, I'm suppose to define a map $$f \colon X \star Y_1 \to X \star Y_2$$ such that the composite of the inclusion $X \hookrightarrow X \star Y$ and $f$ is the same as the inclusion $X \hookrightarrow X \star Y_2$.
My only guess is that $p \colon Y_1 \to Y_2$ induces a morphism $$\tilde{p} \colon X \star Y_1 \to X \star Y_2$$ via \begin{align*} \tilde{p}_n \colon (X \star Y_1)_n & \to (X \star Y_2)_n\\ \coprod_{[i]\oplus [j] = [n]} X_i \times (Y_1)_j & \mapsto \coprod_{[i]\oplus [j] = [n]} X_i \times p_j(Y_1)_j \end{align*}
But I don't know why this map satisfy the commutativity condition with respect to the inclusions $X \hookrightarrow \to X \star Y_k, k \in \{1,2\}$.