Example 1

Let $J$ be a Matlis $A$-module. Put $T(M) = \operatorname{Hom}_A(M, J)$ for every $A$-module $M$ of finite length and $T(f) = \operatorname{Hom}_A(f, 1_J)$ for every $A$-linear mapping $f$ between $A$-modules of finite length. Then conditions FD 1) to FD 5) are satisfied. We shall see below (th. 3) that every construction satisfying conditions FD 1) to FD 5) is obtained in this way.