Consider a category $\mathcal{C}$ with binary coproducts, and the following functors:
- The coproduct functor $L: \mathcal{C} \times \mathcal{C} \rightarrow \mathcal{C}$ defined by $L\left(C, C^{\prime}\right)=C \oplus C^{\prime}$ for every object $\left(C, C^{\prime}\right)$ of $\mathcal{C} \times \mathcal{C}$, and $L\left(f, f^{\prime}\right)=f \oplus f^{\prime}: A \times A^{\prime} \rightarrow B \times B^{\prime}$ for every arrow $\left(f, f^{\prime}\right)$ from $\mathcal{C} \times \mathcal{C}$ with $f: A \rightarrow B$ and $f^{\prime}: A^{\prime} \rightarrow B^{\prime}$.
- The diagonal functor $\Delta: \mathcal{C} \rightarrow \mathcal{C} \times \mathcal{C}$ defined by $\Delta(C)=(C, C)$ for every object $C$ of $\mathcal{C}$, and $\Delta(f)=(f, f):(C, C) \rightarrow\left(C^{\prime}, C^{\prime}\right)$ for every arrow $f: C \rightarrow C^{\prime}$ from $\mathcal{C}$. \
Show that $(L, \Delta)$ is an adjunction.
I found this exercise but I don't know how to start it, someone can help me with an idea.
A morphism $ A \sqcup B \to C$ is the same as a pair of morhisms, one of type $A \to C$ and one of type $B\to C$. Such a pair of morphisms is the same as a morphism $(A,B) \to (C,C)$ in the product category. What I have just described is a bijection \begin{align} \mathcal C(A \sqcup B,C) = \mathcal C(A,C) \times \mathcal C(B,C) = \mathcal C\times \mathcal C((A,B),\Delta C) \end{align} You only have to check that it is natural.