= Category of partial maps localized at subterminal objects
{title2=$\mathcal C_\Phi$}
Given a finite-product category and a product-closed upward filter $\Phi$ of <subobjects> of its <terminal object>, a morphism $A\to B$ in $\mathcal C_\Phi$ is represented by a map $A\times U\to B$ for $U\in\Phi$, with two representatives identified when they agree after restriction to some $W\in\Phi$ below both domains. This construction preserves finite products and, when $\mathcal C$ is <Cartesian closed>, exponentials.
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