The standard reference is Max Kelly’s 1982 paper [“Structures defined by finite limits in the enriched setting”][1].

Perhaps the most unexpected thing is how well the theory works!



  [1]: http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1982__23_1/CTGDC_1982__23_1_3_0/CTGDC_1982__23_1_3_0.pdf