Teacher: In this task, you are given a context, a subject, a relation, and many options. Based on the context, from the options select the object entity that has the given relation with the subject. Answer with text (not indexes).
Teacher: Now, understand the problem? If you are still confused, see the following example:
Context: Joanne McLeod is a Canadian figure skating coach. She is the skating director at the Champs International Skating Centre of BC (formerly known as the BC Centre of Excellence). Here current and former students include Emanuel Sandhu, Mira Leung, Kevin Reynolds, Jeremy Ten, Nam Nguyen, and many others. In 2012, McLeod became the first level 5 certified figure skating coach in British Columbia., Victor Kraatz, MSC (born April 7, 1971) is a Canadian former ice dancer. In 2003, he and his partner, Shae-Lynn Bourne, became the first North American ice dancers to win a World Championship., Allie Hann-McCurdy (born May 23, 1987 in Nanaimo, British Columbia) is a Canadian ice dancer. McCurdy began skating at age eight and was a singles skater until age 12 when she switched to ice dancing. In 2003 she teamed up with Michael Coreno, with whom she was the 2010 Four Continents silver medalist and the 2008 Canadian bronze medalist. The pair retired in June 2010, to coach at the Gloucester Skating Club., Maikki Uotila - Kraatz ( born 25 February 1977 ) is a Finnish ice dancer . She is a former Finnish national champion with Toni Mattila . She married Victor Kraatz on June 19 , 2004 . The two coach in Vancouver , where they are the ice dancing directors at the BC Centre of Excellence . She and Kraatz have two sons , born September 14 , 2006 and July 10 , 2010 ., Burnaby is a city in British Columbia, Canada, located immediately to the east of Vancouver. It is the third-largest city in British Columbia by population, surpassed only by nearby Surrey and Vancouver., Canada (French: ) is a country in the northern half of North America. Its ten provinces and three territories extend from the Atlantic to the Pacific and northward into the Arctic Ocean, covering , making it the world's second-largest country by total area and the fourth-largest country by land area. Canada's border with the United States is the world's longest land border. The majority of the country has a cold or severely cold winter climate, but southerly areas are warm in summer. Canada is sparsely populated, the majority of its land territory being dominated by forest and tundra and the Rocky Mountains. About four-fifths of the country's population of 36 million people is urbanized and live near the southern border. Its capital is Ottawa, its largest city is Toronto; other major urban areas include Montreal, Vancouver, Calgary, Edmonton, Quebec City, Winnipeg and Hamilton., British Columbia (BC) is the westernmost province of Canada, with a population of more than four million people located between the Pacific Ocean and the Rocky Mountains.
British Columbia is also a component of the Pacific Northwest and the Cascadia bioregion, along with the U.S. states of Idaho, Oregon, Washington and Alaska., The "Champs International Skating Centre of British Columbia" (formerly known as the 'BC Centre of Excellence') is one of two major figure skating training centers in Canada. Located in Burnaby, British Columbia, it is home to many great national and international skaters. The programs there are overseen by a staff, including Joanne McLeod, who coaches 3-time Canadian men's national champion Emanuel Sandhu; Bruno Marcotte, who competed at the 2002 Winter Olympics; Victor Kraatz, the 2003 World Champion in ice dancing, and Maikki Uotila, who was a national champion in Finland. The center operates out of Canlan Ice Sports Burnaby 8 Rinks. Notable skaters who train there include Emanuel Sandhu, Mira Leung, Allie Hann-McCurdy & Michael Coreno, Jessica Millar & Ian Moram, Jeremy Ten, and Kevin Reynolds. This skating school is sometimes known as a training site for international competitors to practice for competitions in Vancouver. Champs International hosts its annual competition known as the BC/YK SummerSkate Competition every August., Shae-Lynn Bourne, MSC (born January 24, 1976) is a Canadian ice dancer. In 2003, she and partner Victor Kraatz became the first North American ice dancers to win a World Championship. They competed at three Winter Olympic Games, placing 10th at the 1994 Winter Olympics, 4th at the 1998 Winter Olympics, and 4th at the 2002 Winter Olympics., Vancouver, officially the City of Vancouver, is a coastal seaport city on the mainland of British Columbia, Canada, and the most populous city in the province., Subject: maikki uotila, Relation: country_of_citizenship, Options: (A) american (B) british (C) canada (D) finland (E) montreal
Solution: finland
Reason: This is a good example, as maikki uotila is citizen of the finland.

Now, solve this instance: Context: Given a category C and a morphism f \ colon X \ to Y in C , the image of f is a monomorphism h \ colon I \ to Y satisfying the following universal property : There exists a morphism g \ colon X \ to I such that f = hg . For any object Z with a morphism k \ colon X \ to Z and a monomorphism l \ colon Z \ to Y such that f = lk , there exists a unique morphism m \ colon I \ to Z such that h = lm. Remarks : such a factorization does not necessarily exist g is unique by definition of monic ( = left invertible , abstraction of injectivity ) m is monic . h = lm already implies that m is unique . k = mg The image of f is often denoted by im f or Im ( f ) . One can show that a morphism f is epic if and only if f = im f., In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from "X" to "Y" is often denoted with the notation formula_1., In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word "homomorphism" comes from the ancient Greek language: " (homos)" meaning "same" and " (morphe)" meaning "form" or "shape"., Category theory formalizes mathematical structure and its concepts in terms of a collection of "objects" and of "arrows" (also called morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups., Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics., In various branches of mathematics, a useful construction is often viewed as the most efficient solution to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly., In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term "abstract algebra" was coined in the early 20th century to distinguish this area of study from the other parts of algebra., Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study, in universal algebra one takes "the theory of groups" as an object of study., Subject: image , Relation: part_of, Options: (A) 20th century (B) abstract algebra (C) algebra (D) category theory (E) division (F) learning (G) mathematics (H) theory (I) universal algebra (J) word
Student:
category theory