Internally KK-like spaces and internal inverse limits

We establish equivalences between compacta that admit mappings that limit to the identity, and compacta that are inverse limits of the images under these maps. Our results have relationships to Mardešić and Segalʼs equivalence between polyhedra-like compacta and inverse limits of polyhedra, to the Anderson–Choquet Embedding Theorem, to approximative absolute neighborhood retracts, and to continua that are approximated from within as defined by C.A. Eberhart and J.B. Fugate.