Categorical Pullbacks

The main purpose of this article is to introduce the categorical concept of pullback in Mizar. In the first part of this article we redefine homsets, monomorphisms, epimorpshisms and isomorphisms [6] within a free-object category [15] and it is shown there that ordinal numbers can be considered as categories. Then the pullback is introduced in terms of its universal property, and the Pullback Lemma is formalized. In the last part of the article we formalize the pullback of functors [13] and it is also shown that it is not possible to write an equivalent definition in the context of the previous Mizar formalization of category theory [7].