In his lifetime Unzueta received contemporary cultural recognition through public honors, media coverage, and high-profile international collaborations and commissions.

[5] In 1972 at the age of ten Unzueta was gifted a sketch pad and tempera paint set by his father, which motivated him to further his early creative development.

In 1992 Unzueta reached the age to where he was no longer required to remain in Cuba in the event of mandatory enlistment into the military and was permitted to leave the island.

[2] Seeking to flee the conditions under the Castro regime, he then went into exile in his father's native Spain where he became a part of the international Cuban diaspora.

In 2000 he exhibited his art in the Madrid International Book Fair as a reflection of his established roots as a widely published illustrator dating back to his career in Cuba.

[12][13] These exhibitions in South Florida raised his profile and laid the foundation to facilitate the 2004 opening of Unzueta Gallery in Miami's Little Havana Arts District.

[11] The gallery would be managed by his sister Enaida and would serve as his main base of representation in the United States for the next thirteen years.

[11] The gallery would become a significant contributor to its monthly Viernes Culturales (Cultural Fridays) festival and a hub of artists who would also contribute to its early development of the Little Havana Arts District.

In 2006 Unzueta received two public commissions: the first to create illustrations for the international literary magazine Limon Partido and the second to design the official event poster for Miami-Dade County's Hispanic Heritage Month.

[8][22] The film captured his creative process in making a single work dedicated to Spanish poet Federico García Lorca.

[17] The thieves took advantage of the disorder following the recent impact of Hurricane Katrina and stole twelve original oil on canvas works by Unzueta from several of his prominent series.

[13][21] In addition to his fine arts career, Unzueta also illustrated books, magazine articles, and event posters in Cuba, Spain, and Miami.

[13][24] Unzueta's Magic realism style led to his being commissioned to produce illustrations for several children's storybooks including: Perfume of Violets, Rabbits are Blue, When Crabs Sting, An Important Day, With my Grandma, and The Messenger Star.

[25] In 1999 Unzueta illustrated the first book of short stories by renown Cuban writer, Zoé Valdés, Los Aretes de la Luna, published by Editorial Everest.

[27][23] Unzueta's work has been exhibited alongside other leading Cuban artists such as René Portocarrero, Amelia Peláez, and José Mijares and included in several prominent collections of fine art.

[20] His art career has been covered by various media outlets including El Nuevo Herald, Diario Las Americas, Miami New Times, and Artburst.

[6] The exhibition was positively received by critics and media coverage, including Diario Las Americas, Martí Noticias, and Artburst Miami.