There is a known construction of pushout of $C^*$-algebras, or rather, the amalgamated free product, which is universal for commutative squares of $*$-homomorphisms. Jensen and Thomsen in their book Elements of KK-theory give, in an appendix, a detailed treatment for the free product (i.e the coproduct), for instance. However, $*$-homomorphisms are not the only morphisms between $C^*$-algebras that are reasonable to consider. For instance, a non-proper map $Xto Y$ between locally compact Hausdorff spaces induces a $*$-homomorphism $C_0(Y) to M(C_0(X)) simeq C_b(X)$, which is continuous for the strict topology on $C_0(Y)$. This motivates somewhat the more general type of morphism, where we can take a map $Arightsquigarrow B$ between $C^*$-algebras to be a strictly continuous $*$-homomorphism $Ato M(B)$, where $M(B)$ is the multiplier algebra of $B$. Any $*$-homomorphism $Ato B$ gives one of these more general morphisms, namely the composite $Ato B hookrightarrow M(B)$. It is a fun fact that as a set $M(B)$ is the completion in the strict topology of $B$, though we regard it as being equipped with both the strict topology and its (Banach) $C^*$-topology. These generalised morphisms compose by using the universal property of the strict completion, and so we have a category.
I’m interested in the pushout, in this category, of an arbitrary generalised morphism and a generalised morphism arising from an injective $*$-homomorphism, in particular the inclusion of a full corner.
Conjecture: A full corner inclusion pushes out to also give (the generalised morphism arising from) a full corner inclusion.