Understanding Direct Sums and Product (Vector Spaces) On p.21 of A Course in Homological Algebra - Hilton and Stammbach. there is a proof for a (possibly infinite) collection of modules that there is an isomorphism $\eta$
$\eta: \text{Hom} (\bigoplus_{j \in J} A_j, B) \to \prod_{j \in J} \text{Hom} (A_i, B)$
For $\psi: \bigoplus A_j \to  B$ define  $\eta(\psi) = (\psi i_j)_{j \in J}$
where the $i_j$ are the injections  $i_j: A_j \to \bigoplus A_j$
I'm fine with seeing this as a bijection, and with its linearity.
The proof continues to describe the projections associated with the product, saying the projections $\pi_j : \prod_{j \in J} \text{Hom} (A_i, B) \to \text{Hom} (A_i, B)$
are given by $\pi_j \eta = \text{Hom} (i_j, B) $
I don't see this last part at all (it appears to equate a map with a collection of maps), nor do I have an idea what it should be. Assistance would be appreciated.
 A: Recall that $\hom$ is a functor $\mathcal A^{op}\times\mathcal A\to\mathcal Ab$ when $\mathcal A$ is an Abelian category (for arbitrary categories its codomain is $\mathcal Set$), acting by pre- and post composition.
In particular, $\hom(i_j, B)$ is the map $\hom(\bigoplus_kA_k,\,B)\to\hom(A_j,B)$ assigning $f\mapsto f\,i_j$.
On the other hand, for any $f\in\hom(\bigoplus_kA_k,\,B)$, by definition of $\eta$, the $j$th coordinate of $\eta(f)\in\prod_k\hom(A_k,B)$ is the same: $f\,i_j$.
A: Page 16 of the book:

Let $\beta : B_1 \to B_2$ be a homomorphism of $\Lambda$-modules. We can assign to a homomorphism $\varphi : A \to B_1$, the homomorphism $\beta \varphi : A \to B_2$, thus defining a map $$\beta_* = \operatorname{Hom}_\Lambda(A,\beta) : \operatorname{Hom}_\Lambda(A,B_1) \to \operatorname{Hom}_\Lambda(A,B_2).$$
$\dots$
On the other hand, if $\alpha : A_2 \to A_1$ is a $\Lambda$-module homomorphism, then we can assign to every homomorphism $\varphi : A_1 \to B$ the homomorphism $\varphi\alpha : A_2 \to B$, thus defining a map $$\alpha^* = \operatorname{Hom}_\Lambda(\alpha,B) : \operatorname{Hom}_\Lambda(A_1,B) \to \operatorname{Hom}_\Lambda(A_2,B).$$

