External Direct Product of Projection with Canonical Injection

Theorem
Let $$\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_j, \circ_j}\right), \ldots, \left({S_n, \circ_n}\right)$$ be algebraic structures with identities $$e_1, e_2, \ldots, e_j, \ldots, e_n$$ respectively.

Let $$\left({S, \circ}\right) = \prod_{i=1}^n \left({S_i, \circ_i}\right)$$ be the external direct product $$\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_k, \circ_k}\right), \ldots, \left({S_m, \circ_m}\right)$$.

Let $$pr_j: \left({S, \circ}\right) \to \left({S_j, \circ_j}\right)$$ be the $j$th projection from $$\left({S, \circ}\right)$$ to $$\left({S_j, \circ_j}\right)$$.

Let $$in_j: \left({S_j, \circ_j}\right) \to \left({S, \circ}\right)$$ be the canonical injection from $$\left({S_j, \circ_j}\right)$$ to $$\left({S, \circ}\right)$$.

Then $$pr_j \circ in_j = I_{S_j}$$, where $$I_{S_j}$$ is the identity mapping from $$S_j$$ to $$S_j$$.