Remark

Let $(\mathscr{S}_\lambda)_{\lambda \in \mathrm{L}}$ be a family of structures of species $\Sigma$ on the same set A; let $\mathrm{A}_\lambda$ denote the set A endowed with the structure $\mathscr{S}_\lambda$, and let $\mathrm{I}_\lambda$ denote the identity mapping of A into $\mathrm{A}_\lambda$. Let B be the product set $\mathrm{A}^{\mathrm{L}} = \prod_{\lambda \in \mathrm{L}} \mathrm{A}_\lambda$, and let $\Delta$ be the diagonal of this product (Chapter II, § 5, no. 3). Let $h$ be the diagonal mapping of A onto $\Delta$, so that $h(x)$ is the element $(x_\lambda)_{\lambda \in \mathrm{L}}$ such that $x_\lambda = x$ for all $\lambda \in \mathrm{L}$. Suppose that there exists on B a product structure $\mathscr{S}'$ of the family $(\mathscr{S}_\lambda)$. Since $h$ is injective, criterion CST 15 shows that there exists an initial structure $\mathscr{S}$ with respect to the family $(\mathrm{A}_\lambda, \mathscr{S}_\lambda, \mathrm{I}_\lambda)_{\lambda \in \mathrm{L}}$ if and only if there exists a structure $\mathscr{S}''$ on $\Delta$ induced by $\mathscr{S}'$; $\mathscr{S}''$ is then identical with the structure obtained by transporting $\mathscr{S}$ to $\Delta$ by means of $h$. In particular, when all the structures $\mathscr{S}_\lambda$ are identical, $h$ is an isomorphism of A (endowed with this structure) onto $\Delta$.

We have also the following criterion :

CST16. Let $(\mathrm{A}_\iota)_{\iota \in \mathrm{I}}$, $(\mathrm{B}_\iota)_{\iota \in \mathrm{I}}$ be two families of sets indexed by the same set. For each $\iota \in \mathrm{I}$, let $\mathscr{S}_\iota$ be a structure of species $\Sigma$ on $\mathrm{A}_\iota$ and let $\mathscr{S}'_\iota$ be a structure of species $\Sigma$ on $\mathrm{B}_\iota$. Suppose that there exists on $\mathrm{A} = \prod_{\iota \in \mathrm{I}} \mathrm{A}_\iota \left( \text{resp. on } \mathrm{B} = \prod_{\iota \in \mathrm{I}} \mathrm{B}_\iota \right)$ a product structure $\mathscr{S}$ (resp. $\mathscr{S}'$) of the family $(\mathscr{S}_\iota)$ (resp. $(\mathscr{S}'_\iota)$). For each $\iota \in \mathrm{I}$, let $f_\iota$ be a morphism of $\mathrm{A}_\iota$ into $\mathrm{B}_\iota$. Then the mapping $f = (f_\iota)_{\iota \in \mathrm{I}}$ is a morphism of A into B.

Let $p_\iota$ (resp. $q_\iota$) be the projection of A onto $\mathrm{A}_\iota$ (resp. of B onto $\mathrm{B}_\iota$). Then we have $q_\iota \circ f = f_\iota \circ p_\iota$. Since $f_\iota$ and $p_\iota$ are morphisms (criterion

CST9), $f_i \circ p_i$ is a morphism by (MO$_\Pi$); hence $f$ is a morphism by (IN).