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In this note, we combine the work of Ilmanen and of Colding-Ilmanen-Minicozzito observe a uniqueness property for tangent flows at the first singular timeof a smooth mean curvature flow of a closed surface in 3-dimensional Euclideanspace. Specifically, if, at a fixed singular point, one tangent flow is apositive integer multiple of a shrinking plane, cylinder or sphere, then,modulo rotations, all tangent flows at the point are the same.

A Remark on a Uniqueness Property of High Multiplicity Tangent Flows in Dimension Three