  In this note we remark on the problem of equality of objects in categories
formalized in Martin-L\"of's constructive type theory. A standard notion of
category in this system is E-category, where no such equality is specified. The
main observation here is that there is no general extension of E-categories to
categories with equality on objects, unless the principle Uniqueness of
Identity Proofs (UIP) holds. We also introduce the notion of an H-category, a
variant of category with equality on objects, which makes it easy to compare to
the notion of univalent category proposed for Univalent Type Theory by Ahrens,
Kapulkin and Shulman.
