We give a refinement of the local class field theory of Serre and Hazewinkel. This refinement allows the theory to treat extensions that are not necessarily totally ramified. Such a refinement was obtained and used in the authors' paper on Fontaine's property (Pm), where the explanation had to be rather brief. In this paper, we give a complete account, from necessary knowledge of an appropriate Grothendieck site to the details of the proof. We start by reviewing the local class field theory of Serre and Hazewinkel.Comment: 29 pages, final version to appear in Algebraic Number Theory and Related Topics 2010, 163--191, RIMS K\^oky\^uroku Bessatsu, B32, Res. Inst. Math. Sci. (RIMS), Kyoto, 201
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