If is a category then an ind-object is a strict ind-object (alias essentially monomorphic ind-object) if it can be represented in as (the apex of) a colimit of a small filtered diagram (whose objects are in and) whose morphisms are specifically monomorphisms in .
(Grothendieck-Verdier 71, Exposé I.§8.12.1). See also Blanc 96, def. 4.1.
Dually, strict pro-objects (alias essentially epimorphic pro-objects) are limits of small cofiltered diagrams involving only epimorphisms.
Alexandre Grothendieck, Jean-Louis Verdier, Prefaisceaux, in Theorie de Topos et Cohomologie Etale de Schemas 1971
David Blanc, Colimits for the pro-category of towers of simplicial sets, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1996) Volume: 37, Issue: 4, page 258-278 (numdam)
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