Lax pullback complements and pullbacks of spans

The formation of the "strict" span category Span(C) of a category C with pullbacks is a standard organizational tool of category theory. Unfortunately, limits or colimits in Span(C) are not easily computed in terms of constructions in C. This paper shows how to form the pullback in Span(C) for many, but not all, pairs of spans, given the existence of some specific so-called lax pullback complements in C of the "left legs" of at lea.st one of the two given spans. For some types of spans we require the ambient category to be adhesive to be able to form at least a weak pullback in Span(C). The existence of all lax pullback complements in C along a given morphism is equivalent to the exponentiability of that morphism. Since exponentiability is a rather restrictive property of a morphism, the paper first develops a comprehensive framework of rules for individual lax pullback complement diagrams, which resembles the set of pasting and cancellation rules for pullback diagrams, including their behaviour under pullback. We also present examples of lax pullback complements along non-exponentiable morphisms, obtained via lifting along a fibration.