Injectivity in a category: an overview on smallness conditions

Some of the so called smallness conditions in algebra as well as in category theory, are important and interesting for their own and also tightly related to injectivity, are essential boundedness, cogenerating set, and residual smallness. In this overview paper, we first try to refresh these smallness condition by giving the detailed proofs of the results mainly by Bernhard Banaschewski and Walter Tholen, who studied these notions in a much more categorical setting. Then, we study these notions as well as the well behavior of injectivity, in the class mod(Sigma,mathcalE) of models of a set Sigma of equations in a suitable category, say a Grothendieck topos mathcalE , given by M.Mehdi Ebrahimi. We close the paper by some examples to support the results.