Math friends: Whats a good name for maps f:(X,x)->(Y,y) of - TopicsExpress

Math friends: Whats a good name for maps f:(X,x)->(Y,y) of pointed spaces such that x is the only point p such that f(p)=y? Id be very happy to learn the widely agreed-upon name, but I doubt that there is one. Id also be happy with an inspired suggestion, or even a sensible suggestion. My context is PL topology, so these are the pointed PL maps that induce a map from the link of x in X to the link of y in Y.

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