Proposition 4.7.4. Let $\seqi{X}$, $\seqi{Y}$ be topological/uniform spaces and $\seqi{f}$ such that $f_{i} \in C(X_{i}; Y_{i})$ is an embedding for all $i \in I$, then
There exists a unique $f \in C(\prod_{i \in I}X_{i}; \prod_{i \in I}Y_{i})$/$f \in UC(\prod_{i \in I}X_{i}; \prod_{i \in I}Y_{i})$ such that the following diagram commutes
\[\xymatrix{ \prod_{i \in I}Y_i \ar@{->}[r]^{\pi_i} & Y_i \\ \prod_{i \in I} X_i \ar@{->}[r]_{\pi_i} \ar@{->}[u]^{f} & X_i \ar@{^{(}->}[u]_{f_i} }\]for all $i \in I$.
$f$ is an embedding.
Proof. (1): By (U) of Definition 4.7.1/Definition 5.2.5.
(2): Consider the following diagram
\[
% https://darknmt.github.io/res/xypic-editor/#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
\xymatrix{ \prod_{i \in I}Y_i \ar@{->}[rd]^{\pi_i} & \\ \iota_P(\prod_{i \in I}X_i) \ar@{->}[u] & Y_i \\ \prod_{i \in I} X_i \ar@{->}[r]_{\pi_i} \ar@{->}[u]^{f} & X_i \ar@{^{(}->}[u]_{f_i} }\]
Since each $X_{i} \to Y_{i}$ is an embedding, the composition
\[\xymatrix{ \iota_P(\prod_{i \in I}X_i) \ar@{->}[r]^{\pi_i} & Y_i \ar@{->}[r]^{f_i^{-1}} & X_i }\]
is continuous/uniformly continuous. By (U) of Definition 4.7.1/Definition 5.2.5, $f$ is an embedding.$\square$