Pseudo-compactness of direct limits

Let Z, H be spaces. We introduce the direct system X induced by the set of maps between the spaces Z and H and explore conditions under which X = dirlim X will be pseudo-compact, but not necessarily compact. We also define the concept of a space Y being an absolute extensor relative to convergent sequences for a given space Z, which is useful in our situation and is used in formulating sufficient conditions for X = dirlim X to be pseudo-compact.