The paper deals with closed categories over complete lattice-ordered monoids \((L, \vee, \wedge, \ast, 1)\). Covariant and contravariant fuzzy Galois connections were introduced and examined by \textit{R. Bělohlávek} [Math. Log. Q. 45, No.~4, 497--504 (1999; Zbl 0938.03079)], and \textit{G. Georgescu} and \textit{A. Popescu} [Soft Comput. 7, No.~7, 458--467 (2003; Zbl 1024.03025)] in the case of commutative semigroups. This paper examines the case of non-commutative semigroups. It describes links and differences between the commutative and the non-commutative cases in the terminology of categories enriched over \(L\).