Sentence:  Given a homotopical category X and a <WSD>functor</WSD> u: A \rightarrow B, a homotopical u-colimit (resp. u-limit) <WSD>functor</WSD> on X will be a homotopically terminal (resp. initial) Kan extension of the identity (50.2) along the induced diagram <WSD>functor</WSD> X^u: X^B \rightarrow X^A (47.1). 
Meaning of the word marked with <WSD> tags:  A category homomorphism; a morphism from a source category to a target category which maps objects to objects and arrows to arrows, in such a way as to preserve domains and codomains (of the arrows) as well as composition and identities.