This may be a simple piece of terminology, but I have not located it. For the initial topology, we are given a set of functions, indexed by $\alpha$, $f_{\alpha}:X\rightarrow Y_{\alpha}$, where each codomain $Y_{\alpha}$ is a topological space. The initial topology is the coarses topology making all the $f_{\alpha}$ continuous, and absent any additional information, this is often the most natural topology to use on $X$.

However, what about when the space $X$ I want to topologize is not the domain of the functions $f_{\alpha}$? Instead, each domain $\tilde{X}_{\alpha}$ is constructed out of $X$ and other building blocks. For a simple example, if $Z$ is locally compact Hausdorff, the coarsest topology on the continuous function space $X=C(Z,Y)$ that makes the evaluation map $e:C(Z,Y)\times Z\rightarrow Y$ defined by $e(f,z)=f(z)$ continuous is the compact-open topology. Here $X$ was not the domain but rather a factor in the domain, so this is not just the initial topology.

Is there a name for this construction? Ideally, the term would apply to more general situations in which the $\tilde{X}_{\alpha}$ are constructed out of $X$ and other spaces with known topologies—not just in situations in which the domains each contain $X$ as a factor. (And, although it is not directly relevant to anything I'm doing, is there then also a name for a corresponding generalization of the final topology?)