Category of partial maps localized at subterminal objects (${\mathcal{C}}_{\Phi }$)

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.