  We give an introduction to constructive category theory by answering two
guiding computational questions. The first question is: how do we compute the
set of all natural transformations between two finitely presented functors like
$\mathrm{Ext}$ and $\mathrm{Tor}$ over a commutative coherent ring $R$? We give
an answer by introducing category constructors that enable us to build up a
category which is both suited for performing explicit calculations and
equivalent to the category of all finitely presented functors. The second
question is: how do we determine the differentials on the pages of a spectral
sequence associated to a filtered cochain complex only in terms of operations
directly provided by the axioms of an abelian category? Its answer relies on a
constructive method for performing diagram chases based on a calculus of
relations within an arbitrary abelian category.
