@MISC{CAT_8.MIZ,
  AUTHOR = {Riccardi, Marco},
  TITLE = {Exponential Objects},
  SECTION1 = {Preliminaries},
  SECTION2 = {Terminal Objects},
  SECTION3 = {Initial Objects},
  SECTION4 = {Categorical Products},
  SECTION5 = {Natural Transformations},
  SECTION6 = {Exponential Objects},
  DAY = {15},
  MONTH = {August},
  YEAR = {2015},
  ADDRESS1 = {Via del Pero 102\\54038 Montignoso\\Italy},
  SUMMARY = {In the first part of this article we formalize the concepts of terminal and initial
  object, categorical product \cite{Borceaux} and natural transformation within a free-object
  category \cite{Adamek:2009}. In particular, we show that this definition of natural transformation
  is equivalent to the standard definition \cite{MacLane:1}. Then we introduce the exponential
  object using its universal property and we show the isomorphism between the exponential object
  of categories and the functor category\cite{Lawvere:1963}.}}

@BOOK{Adamek:2009,
       AUTHOR = {Adamek, Jiri and Herrlich, Horst and Strecker, George E.},
        TITLE = {Abstract and Concrete Categories: The Joy of Cats},
    PUBLISHER = {Dover Publication},
         YEAR = {2009},
      ADDRESS = {New York}}

@BOOK{Borceaux,
       AUTHOR = {Borceaux, Francis},
        TITLE = {Handbook of Categorical Algebra I. {B}asic Category Theory},
    PUBLISHER = {Cambridge University Press},
         YEAR = {1994},
       VOLUME = {50},
       SERIES = {Encyclopedia of Mathematics and its Applications},
      ADDRESS = {Cambridge}}

@BOOK{MacLane:1,
       AUTHOR = {Mac Lane, Saunders},
        TITLE = {Categories  for the Working Mathematician},
    PUBLISHER = {Springer Verlag},
         YEAR = {1971},
       VOLUME = {5},
       SERIES = {Graduate Texts in Mathematics},
      ADDRESS = {New York, Heidelberg, Berlin}}

@ARTICLE{Lawvere:1963,
       AUTHOR = {Lawvere, F. William},
        TITLE = {Functorial Semantics of Algebraic Theories and
                 Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories},
      JOURNAL = {Reprints in Theory and Applications of Categories},
         YEAR = {2004},
       VOLUME = {5},
        PAGES = {1--121}}
