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The Baues--Wirsching cohomology is one of the cohomologies of a smallcategory. Our aim is to describe the first Baues--Wirsching cohomology of thesmall category generated by a finite quiver freely. We consider the case wherethe coefficient is a natural system obtained by the composition of a functorand the target functor. We give an algorithm to obtain generators of the vectorspace of inner derivations. It is known that there exists a surjection from thevector space of derivations of the small category to the first Baues--Wirschingcohomology whose kernel is the vector space of inner derivations.

On computation of the first Baues--Wirsching cohomology of a freely-generated small category