Let $\{X_{\alpha} \}_{\alpha \in A}$ be a collection of Banach spaces. It is easy to show that $ P = \{(x_{\alpha}) : {\rm sup}_{\alpha} \|x_{\alpha} \| < \infty \} $ with $\| (x_{\alpha} ) \| = {\rm sup}_{\alpha} \| x_{\alpha} \|$ is a banach space.

If the indexing set $A$ is finite, then it is easy to show that $P$ (with the natural projection maps) is the product of $\{X_{\alpha} \}_{\alpha \in A}$ in the category of banach spaces.

Assume $Y$ is a banach space and $f_{\alpha} : Y \to X_{\alpha}$ is a linear continuous map for each $\alpha$. If ${\rm sup}_{\alpha} \|f_{\alpha} \| < \infty$ then it is easy to see that the induced linear map $Y \to \Pi X_{\alpha}$ has image a subset of $P$ and is continuous. If this condition is not satisfied then it is not clear to me that there exists a continuous linear map $g:Y \to P$ such that $\pi_{\alpha} \circ g = f_{\alpha}$ for all $\alpha$.

Is there any way to prove that $P$ is the categorical product of the collection $\{X_{\alpha} \}_{\alpha \in A}$ when $A$ is infinite?