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This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mrow></mrow><mrow><mn>0</mn></mrow><mi>c</mi></msubsup><msubsup><mi>D</mi><mrow><msup><mn>0</mn><mo>+</mo></msup></mrow><mi>q</mi></msubsup><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mi>λ</mi><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mo>⊕</mo><mi>f</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>u</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo>⊕</mo><mi>B</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mi>C</mi><mrow><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><mi>t</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow></mrow></semantics></math></inline-formula> with initial conditions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mo>=</mo><msub><mi>u</mi><mn>0</mn></msub><mo>,</mo><msup><mi>u</mi><mo>′</mo></msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mo>=</mo><msub><mi>u</mi><mn>1</mn></msub><mo>.</mo></mrow></semantics></math></inline-formula> Moreover, by using direct analytic methods, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mi>q</mi></msub></semantics></math></inline-formula>–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.