170 VIII. HOMOTOPY COLIMIT AND LIMIT FUNCTORS AND HOMOTOPICAL ONES while 49.1 and 51.4(ii)/ imply that (ii) a sufficient condition for the homotopical cocompleteness (resp. complete- ness) of X, i.e. the existence of homotopical colimit (resp. limit) systems on X, is that (ii)' there exists a terminal (resp. initial) Kan extension along X^ (resp. X^ ) which is left (resp. right) deformable (48.6) Moreover it follows from 49.1 and 51.4(i) and (iii) that, if X is cocomplete (resp. complete), then (iii) there exist homotopical colimit (resp. limit) systems on X iff (iii)' there exist homotopy colimit (resp. limit) systems on X (49.2) and (iii)" if (iii) and (iii)' hold then the homotopy colimit (resp. limit) systems on X are exactly the colinrcat'- (resp. Ymr-cat'-) presentations (b\A(\v)) of the homotopical colimit (resp. limit) systems on X so that (iv) for cocomplete (resp. complete) homotopical categories the above notion of homotopical cocompleteness (resp. completeness) coincides with the one we considered in §49.
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